1. MIT

   Can a \(2018\times 2018\) grid be tiled with rotations and reflections of an “L”-shaped tetromino?

2. MIT

   Choose any \(1000\) points in the plane. Does there always exist a straight line which divides the points exactly in half? I.e. such that exactly \(500\) of the points lie on one side of the line and \(500\) on the other side?

3. MIT

   A \(6 \times 6\) grid is tiled by dominoes. Prove that there exists some line which cuts the board into two nonempty parts without cutting through any domino.

4. MIT

   Let \(B\) be a set of a more than \(2^{n+1}/n\) distinct points with coordinates of the form \((\pm 1, \pm 1, \dotsc, \pm 1)\) in \(n\)-dimensional space for some \(n \gt 2.\) Show that there are three distinct points in \(B\) which are the vertices of an equilateral triangle.

5. Putnam & Beyond

   An equilateral triangle is divided into \(n^2\) equal equilateral triangles by lines parallel to the sides. We call a chain a sequence of triangles these smaller triangles in which no triangle appears twice and every two consecutive triangles share a side. What is the largest possible number of triangles in a chain?

6. Putnam & Beyond

   An equilateral triangle is divided into \(n^2\) equal equilateral triangles by lines parallel to the sides. From among the vertices of the triangles obtained this way, \(m\) vertices are chosen such that such that for any two of the chosen vertices, the segment connecting them is not parallel to any of the sides of the original triangle. What is the largest possible value that \(m\) may be?

7. Putnam & Beyond

   In a regular \(2n\)-gon, \(n\) diagonals intersect at a point \(S,\) which is not a vertex. Prove that \(S\) is the center of the \(2n\)-gon.

8. Putnam & Beyond

   Prove that any \(n\) points in the plane can be covered by finitely many disks with the sum of the diameters of the disks less than \(n\) and the distance between any two disks greater than 1.

9. Putnam 2017 A6

   The \(30\) edges of a regular icosahedron are distinguished by labeling them \(1,2,\dotsc,30.\) How many different ways are there to paint each edge red, white, or blue such that each of the \(20\) triangular faces of the icosahedron has two edges of the same color and a third edge of a different color?